Although not significant, the central value of the branching fraction of
\(\psi (3770)\rightarrow p\bar{p}\pi ^0\)  is also extracted to be either
\((1.1\pm 5.4\pm 0.1\pm 0.1^{\Gamma _{e^+e^-}})\times 10^{-6}\)  or
\((0.33\pm 0.01\pm 0.03\pm 0.03^{\Gamma _{e^+e^-}})\) %,
where the errors are statistical, systematic, and from the uncertainty
of quoted \(\Gamma _{e^+e^-}\)  [1]}.
The two solutions suggest
either noticeable or negligible charmless decays of \(\psi (3770)\rightarrow p\bar{p}\pi ^0\) .
As one of the charmless decays of the \(\psi (3770)\) , the branching fraction can
also be estimated considering the mixing with the \(\psi (3686)\) 
due to tensor forces and coupling to charmed meson pairs [2]}.
This S- and D-wave charmonium mixing model accounts for the “\(\rho \pi \)  puzzle”
by converting expected \(\psi (3686)\rightarrow \rho \pi \)  to corresponding \(\psi (3770)\)  partial widths.
Taking the \(J/\psi \)  and the \(\psi (3686)\)  decays into the same final state as input,
the branching fraction of \(\psi (3770)\rightarrow p\bar{p}\pi ^0\)  is predicted in the range
[\((2.9\pm 4.9)\times 10^{-7}\) , \((15.3\pm 1.5)\times 10^{-5}\) ] assuming the mixing angle
\(\theta =(12\pm 2)^{\circ }\) . One of our fit results falls
in the range of this prediction, while the other solution is beyond
the range by one order of magnitude.
If we use another mixing angle \(\theta =-(27\pm 2)^{\circ }\) ,
the model predicts the branching fraction to be in the range
[\((2.4\pm 1.9)\times 10^{-7}\) , \((2.9\pm 0.3)\times 10^{-5}\) ].
Compared to the results with \(\theta =12^{\circ }\) , the range
becomes narrower, and the maximum decay rate of \(\psi (3770)\rightarrow p\bar{p}\pi ^0\) 
is much smaller.
